Compact Orientable Manifold Research Articles

Analytic Detection in Homotopy Groups of Smooth Manifolds

In this paper, for the mapping of a sphere into a compact orientable manifold Sn → M, n ≥ 1, we solve the problem of determining whether it represents a nontrivial element in the homotopy group of the manifold πn(M). For this purpose, we consistently use the theory of iterated integrals developed by Chen. It should be noted that the iterated integrals as repeated integration were previously meaningfully used by Lappo-Danilevsky to represent solutions of systems of linear differential equations and by Whitehead for the analytical description of the Hopf invariant for mappings f : S2n−1 → Sn, n ≥ 2.We give a brief description of Chen’s theory representing Whitehead’s and Haefliger’s formulas for the Hopf invariant and generalized Hopf invariant. Examples of calculating these invariants using the technique of iterated integrals are given. Further, it is shown how one can detect any element of the fundamental group of a Riemann surface using iterated integrals of holomorphic forms. This required to prove that the intersection of the terms of the lower central series of the fundamental group of a Riemann surface is a unit group.

Simple numerical solutions to the Einstein constraints on various three-manifolds

Numerical solutions to the Einstein constraint equations are constructed on a selection of compact orientable three-dimensional manifolds with non-trivial topologies. A simple constant mean curvature solution and a somewhat more complicated non-constant mean curvature solution are computed on example manifolds from three of the eight Thursten geometrization classes. The constant mean curvature solutions found here are also solutions to the Yamabe problem that transforms a geometry into one with constant scalar curvature.

A necessary and sufficient condition for a Heegaard splitting to be keen weakly reducible

In this work, we first present a necessary and sufficient condition to closed orientable 3-manifold admitting keen weakly reducible Heegaard splitting. By an example, we show that condition is not suitable for compact orientable manifold with non-sphere boundary component. At last, we give a necessary and sufficient condition for compact orientable 3-manifold to have a keen weakly reducible Heegaard splitting.

Reconstructing the orbit type stratification of a torus action from its equivariant cohomology

We investigate what information on the orbit type stratification of a torus action on a compact space is contained in its rational equivariant cohomology algebra. Regarding the (labelled) poset structure of the stratification, we show that equivariant cohomology encodes the subposet of ramified elements. For equivariantly formal actions, we also examine what cohomological information of the stratification is encoded. In the smooth setting, we show that under certain conditions—which in particular hold for a compact orientable manifold with discrete fixed point set—the equivariant cohomologies of the strata are encoded in the equivariant cohomology of the manifold.

Gauge freedom in magnetostatics and the effect on helicity in toroidal volumes

Magnetostatics defines a class of boundary value problems in which the topology of the domain plays a subtle role. For example, representability of a divergence-free field as the curl of a vector potential comes about because of homological considerations. With this in mind, we study gauge freedom in magnetostatics and its effect on the comparison between magnetic configurations through key quantities such as the magnetic helicity. For this, we apply the Hodge decomposition of k-forms on compact orientable Riemaniann manifolds with smooth boundary, as well as de Rham cohomology, to the representation of magnetic fields through potential one-forms in toroidal volumes. An advantage of the homological approach is the recovery of classical results without explicit coordinates and assumptions about the fields on the exterior of the domain. In particular, a detailed construction of minimal gauges and a formal proof of relative helicity formulas are presented.

Compact hyperbolic manifolds without spin structures

We exhibit the first examples of compact orientable hyperbolic manifolds that do not have any spin structure. We show that such manifolds exist in all dimensions $n \geq 4$. The core of the argument is the construction of a compact orientable hyperbolic $4$-manifold $M$ that contains a surface $S$ of genus $3$ with self intersection $1$. The $4$-manifold $M$ has an odd intersection form and is hence not spin. It is built by carefully assembling some right angled $120$-cells along a pattern inspired by the minimum trisection of $\mathbb{C}\mathbb{P}^2$. The manifold $M$ is also the first example of a compact orientable hyperbolic $4$-manifold satisfying any of these conditions: 1) $H_2(M,\mathbb{Z})$ is not generated by geodesically immersed surfaces. 2) There is a covering $\tilde{M}$ that is a non-trivial bundle over a compact surface.

The syzygy order of big polygon spaces

Big polygon spaces are compact orientable manifolds with a torus action whose equivariant cohomology can be torsion-free or reflexive without being free as a module over $H^*(BT)$. We determine the exact syzygy order of the equivariant cohomology of a big polygon space in terms of the length vector defining it. The proof uses a refined characterization of syzygies in terms of certain linearly independent elements in $H^2(BT)$ adapted to the isotropy groups occurring in a given $T$-space.

On Three Dimensional Cosymplectic Manifolds Admitting Almost Ricci Solitons

In the present paper we study three dimensional cosymplectic manifolds admitting almost Ricci solitons. Among others we prove that in a three dimensional compact orientable cosymplectic manifold M^3 withoutboundary an almost Ricci soliton reduces to Ricci soliton under certain restriction on the potential function lambda. As a consequence we obtain several corollaries. Moreover we study gradient almost Ricci solitons.

Determining hyperbolicity of compact orientable 3-manifolds with torus boundary

Thurston's hyperbolization theorem for Haken manifolds and normal surface theory yield an algorithm to determine whether or not a compact orientable 3-manifold with nonempty boundary consisting of tori admits a complete finite-volume hyperbolic metric on its interior. A conjecture of Gabai, Meyerhoff, and Milley reduces to a computation using this algorithm.

Scalar, vector and tensor harmonics on the flat compact orientable three-manifolds

Observations suggest that our universe is spatially flat on the largest observable scales. Exactly six different compact orientable three-dimensional manifolds admit flat metrics. These six manifolds are therefore the most natural choices for building cosmological models based on the present observations. This paper briefly describes these six manifolds and the harmonic basis functions previously developed for representing arbitrary scalar fields on them. The principal focus of this paper is the development of new harmonics for representing arbitrary vector and second-rank tensor fields on these manifolds. These new harmonics are designed to be useful tools for analyzing the dynamics of electromagnetic and gravitational fields on these spaces.

Topological canal foliations

Regular canal surfaces of $\mathbb{R}^3$ or $\mathbb{S}^3$ admit foliations by circles: the characteristic circles of the envelope. In order to build a foliation of $\mathbb{S}^3$ with leaves being canal surfaces, one has to relax the condition “canal” a little (“weak canal condition”) in order to accept isolated umbilics. Here, we define a topological condition which generalizes this “weak canal” condition imposed on leaves, and classify the foliations of compact orientable 3-manifolds we can obtain this way.

The $$S^1$$ S 1 -Equivariant Signature for Semi-free Actions as an Index Formula

Lott (Math Ann 316(4):617–657, 2000) defined an integer-valued signature $$\sigma _{S^1}(M)$$ for the orbit space of a compact orientable manifold with a semi-free $$S^1$$ -action but he did not construct a Dirac-type operator which has this signature as its index. We construct such operator on the orbit space and we show that it is essentially unique and that its index coincides with Lott’s signature, at least when the stratified space satisfies the so-called Witt condition. For the non-Witt case, this operator remains essentially self-adjoint (in contrast to the Hodge-de Rham operator) and it has a well-defined index which we conjecture will also compute $$\sigma _{S^1}(M)$$ .

Energy identity and necklessness for a sequence of Sacks–Uhlenbeck maps to a sphere

Let u be a map from a Riemann surface M to a Riemannian manifold N and α>1, the α energy functional is defined asEα(u)=12∫M[(1+|▽u|2)α−1]dV.We call uα a sequence of Sacks–Uhlenbeck maps if uα are critical points of Eα andsupα>1⁡Eα(uα)<∞.In this paper, we show the energy identity and necklessness for a sequence of Sacks–Uhlenbeck maps during blowing up, if the target N is a sphere SK−1. The energy identity can be used to give an alternative proof of Perelman's result [15] that the Ricci flow from a compact orientable prime non-aspherical 3-dimensional manifold becomes extinct in finite time (cf. [3,4]).

Szegő kernel asymptotics for high power of CR line bundles and Kodaira embedding theorems on CR manifolds

Let X be an abstract not necessarily compact orientable CR manifold of dimension 2n − 1, n > 2, and let L be the k-th tensor power of a CR complex line bundle L over X. Given q ∈ {0, 1, . . . , n− 1}, let (q) b,k be the Gaffney extension of Kohn Laplacian for (0, q) forms with values in L. For λ ≥ 0, let Π (q) k,≤λ := E((−∞, λ]), where E denotes the spectral measure of (q) b,k . In this work, we prove that Π (q) k,≤k−N0 F ∗ k , FkΠ (q) k,≤k−N0 F ∗ k , N0 ≥ 1, admit asymptotic expansions on the non-degenerate part of the characteristic manifold of (q) b,k , where Fk is some kind of microlocal cut-off function. Moreover, we show that FkΠ (q) k,≤0F ∗ k admits a full asymptotic expansion if (q) b,k has small spectral gap property with respect to Fk and Π (q) k,≤0 is k-negligible away the diagonal with respect to Fk. By using these asymptotics, we establish almost Kodaira embedding theorems on CR manifolds and Kodaira embedding theorems on CR manifolds with transversal CR S1 actions.

Epimorphisms of 3-manifold groups

Let f: M -> N be a proper map between two aspherical compact orientable 3-manifolds with empty or toroidal boundary. We assume that N is not a closed graph manifold. Suppose that f induces an epimorphism on fundamental groups. We show that f is homotopic to a homeomorphism if one of the following holds: either for any finite-index subgroup Gamma of pi(1)(N) the ranks of Gamma and of f(*)(-1) (Gamma) agree, or for any finite cover (N) over tilde of N the Heegaard genus of (N) over tilde and the Heegaard genus of the pull-back cover (M) over tilde agree.

An averaging formula for the coincidence Reidemeister trace

In the setting of continuous maps between compact orientable manifolds of the same dimension, there is a well known averaging formula for the coincidence Lefschetz number in terms of the Lefschetz numbers of lifts to some finite covering space. We state and prove an analogous averaging formula for the coincidence Reidemeister trace. This generalizes a recent formula in fixed point theory by Liu and Zhao. We give two separate and independent proofs of our main result: one using methods developed by Kim and the first author for averaging Nielsen numbers, and one using an axiomatic approach for the local Reidemeister trace. We also give some examples and state some open questions for the nonorientable case.

Minimal 4-colored graphs representing an infinite family of hyperbolic 3-manifolds

The graph complexity of a compact 3-manifold is defined as the minimum order among all 4-colored graphs representing it. Exact calculations of graph complexity have been already performed, through tabulations, for closed orientable manifolds (up to graph complexity 32) and for compact orientable 3-manifolds with toric boundary (up to graph complexity 12) and for infinite families of lens spaces. In this paper we extend to graph complexity 14 the computations for orientable manifolds with toric boundary and we give two-sided bounds for the graph complexity of tetrahedral manifolds. As a consequence, we compute the exact value of this invariant for an infinite family of such manifolds.

On the singularities of the Szegő projections on lower energy forms

Let $X$ be an abstract not necessarily compact orientable CR manifold of dimension $2n-1$, $n\geqslant2$. Let $\Box^{(q)}_{b}$ be the Gaffney extension of Kohn Laplacian for $(0,q)$-forms. We show that the spectral function of $\Box^{(q)}_b$ admits a full asymptotic expansion on the non-degenerate part of the Levi form. As a corollary, we deduce that if $X$ is compact and the Levi form is non-degenerate of constant signature on $X$, then the spectrum of $\Box^{(q)}_b$ in $]0,\infty[$ consists of point eigenvalues of finite multiplicity. Moreover, we show that a certain microlocal conjugation of the associated Szeg\"o kernel admits an asymptotic expansion under a local closed range condition. As applications, we establish the Szeg\"o kernel asymptotic expansions on some weakly pseudoconvex CR manifolds and on CR manifolds with transversal CR $S^1$ actions. By using these asymptotics, we establish some local embedding theorems on CR manifolds and we give an analytic proof of a theorem of Lempert asserting that a compact strictly pseudoconvex CR manifold of dimension three with a transversal CR $S^1$ action can be CR embedded into $\mathbb{C}^N$, for some $N\in\mathbb N$.

$C^1$-Robust Topologically Mixing Solenoid-Like Attractors and Their Invisible Parts

The aim of this paper is to discuss statistical attractors of skew products over the solenoid which have an $m$-dimensional compact orientable manifold $M$ as a fiber and their $\varepsilon$-invisible parts, i.e. a sizable portion of the attractor in which almost all orbits visit it with average frequency no greater than $\varepsilon$.We show that for any $n \in \mathbb{N}$ large enough, there exists a ball $D_n$ in thespace of skew products over the solenoid with the fiber $M$ such that each $C^2$-skewproduct map from $D_n$ possesses a statistical attractor with an $\varepsilon$-invisible part,whose size of invisibility is comparable to that of the whole attractor. Also, itconsists of structurally stable skew product maps.In particular, small perturbations of these skew products in the space of all diffeomorphisms still have attractors with the same properties.\\Our construction develops the example of (Ilyashenko \& Negut, 2010) to skew products over the solenoidwith an $m$-dimensional fiber, $m \geq 2$.As a consequence, we provide a class of local diffeomorphisms acting on$S^1 \times M$ such that each map of this class admits a robustly topologically mixing maximal attractor.

Boundary degeneracy of topological order

We introduce the concept of boundary degeneracy of topologically ordered states on a compact orientable spatial manifold with boundaries, and emphasize that the boundary degeneracy provides richer information than the bulk degeneracy. Beyond the bulk-edge correspondence, we find the ground state degeneracy of the fully gapped edge modes depends on boundary gapping conditions. By associating different types of boundary gapping conditions as different ways of particle or quasiparticle condensations on the boundary, we develop an analytic theory of gapped boundaries. By Chern-Simons theory, this allows us to derive the ground state degeneracy formula in terms of boundary gapping conditions, which encodes more than the fusion algebra of fractionalized quasiparticles. We apply our theory to Kitaev's toric code and Levin-Wen string-net models. We predict that the $Z_2$ toric code and $Z_2$ double-semion model (more generally, the $Z_k$ gauge theory and the $U(1)_k \times U(1)_{-k}$ non-chiral fractional quantum Hall state at even integer $k$) can be numerically and experimentally distinguished, by measuring their boundary degeneracy on an annulus or a cylinder.